Using a small melon baller, scoop the cantaloupe and the honeydew into balls or use a knife to cut melons into cubes. You should have about 4 cups cantaloupe pieces and 2 cups honeydew pieces. Set melon pieces aside.
In a small saucepan combine pineapple juice, sugar, and ginger. Bring to boiling, stirring until sugar dissolves; reduce heat. Simmer, uncovered, over medium heat for 5 to 7 minutes or until the mixture is the consistency of a thin syrup. Remove from heat; cool. Transfer syrup to a storage container. Add 2 cups of the cantaloupe pieces and all of the honeydew pieces. Cover and chill overnight.
Meanwhile, in a large bowl stir together sour cream and yogurt; set aside. In a blender or food processor, blend or process strawberries until smooth; add to sour cream mixture. Cover and blend or process remaining 2 cups cantaloupe pieces until smooth. Add pureed melon and the milk to strawberry mixture; stir to combine. Cover and chill overnight.
To serve, drain melon pieces, reserving syrup. Stir reserved syrup into the chilled soup. Ladle soup into bowls; top with melon pieces. If desired, garnish with mint leaves.